Abstract
Sidorenko’s conjecture states that the number of copies of a bipartite graph H in a graph G is asymptotically minimised when G is a quasirandom graph. A notorious example where this conjecture remains open is when H = K5,5C10. It was even unknown whether this graph possesses the strictly stronger, weakly norming property.
We take a step towards understanding the graph K5,5C10 by proving that it is not weakly norming. More generally, we show that ‘twisted’ blow-ups of cycles, which include K5,5C10 and C6☐K2, are not weakly norming. This answers two questions of Hatami. The method relies on the analysis of Hessian matrices defined by graph homomorphisms, by using the equivalence between the (weakly) norming property and convexity of graph homomorphism densities. We also prove that Kt,t minus a perfect matching, proven to be weakly norming by Lovász, is not norming for every t > 3.
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References
S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
D. Conlon, J. Fox and B. Sudakov, An approximate version of Sidorenko’s conjecture, Geometric and Functional Analysis 20 (2010), 1354–1366.
D. Conlon, J. H. Kim, C. Lee and J. Lee, Some advances on Sidorenko’s conjecture, Journal of the London Mathematical Society 98 (2018), 593–608.
D. Conlon and J. Lee, Finite reflection groups and graph norms, Advances in Mathematics 315 (2017), 130–165.
D. Conlon and J. Lee, Sidorenko’s conjecture for blow-ups, Discrete Analysis, to appear, https://arxiv.org/abs/1809.01259.
M. Doležal, J. Grebík, J. Hladký, I. Rocha and V. Rozhoň, Cut distance identifying graphon parameters over weak* limits, https://arxiv.org/abs/1809.03797v3.
P. Erdős and M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181–192.
F. Garbe, J. Hladký and J. Lee, Two remarks on graph norms, Discrete & Computational Geometry, to appear, https://arxiv.org/abs/1909.10987.
H. Hatami, Graph norms and Sidorenko’s conjecture, Israel Journal of Mathematics 175 (2010), 125–150.
J. H. Kim, C. Lee and J. Lee, Two approaches to Sidorenko’s conjecture, Transactions of the American Mathematical Society 368 (2016), 5057–5074.
D. Král’, T. Martins, P. P. Pach and M. Wrochna, The step Sidorenko property and non-norming edge-transitive graphs, Journal of Combinatorial Theory. Series A 162 (2019), 34–54.
J. Li and B. Szegedy, On the logarithmic calculus and Sidorenko’s conjecture, Combinatorica, to appear.
L. Lovász, Graph homomorphisms: Open problems, https://web.cs.elte.hu/∼lovasz/problems.pdf.
L. Lovász, Large Networks and Graph Limits, American Mathematical Society Colloquium Publications, Vol. 60, American Mathematical Society, Providence, RI, 2012.
L. Lovász and B. Szegedy, Limits of dense graph sequences, Journal of Combinatorial Theory. Series B 96 (2006), 933–957.
A. Sidorenko, Inequalities for functionals generated by bipartite graphs, Discrete Mathematics and Applications 2 (1992), 489–504.
A. Sidorenko, A correlation inequality for bipartite graphs, Graphs and Combinatorics 9 (1993), 201–204.
B. Szegedy, An information theoretic approach to Sidorenko’s conjecture, https://arxiv.org/abs/1406.6738.
Acknowledgements
We would like to thank Sasha Sidorenko for suggesting to prove Theorem 1.6, and Jan Hladký for explaining the result [6] in which he and his colleagues obtained a partial version of Theorem 1.5. We are also grateful to David Conlon, Christian Reiher, and Mathias Schacht for helpful comments and discussions.
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Research supported by ERC Consolidator Grant PEPCo 724903.
Research supported by G.I.F. Grant Agreements No. I-1358-304.6/2016.
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Lee, J., Schülke, B. Convex graphon parameters and graph norms. Isr. J. Math. 242, 549–563 (2021). https://doi.org/10.1007/s11856-021-2112-6
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DOI: https://doi.org/10.1007/s11856-021-2112-6